A051064 -id:A051064 - cp's OEIS frontend

A004128 a(n) = Sum_{k=1..n} floor(3*n/3^k).

0, 1, 2, 4, 5, 6, 8, 9, 10, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 30, 31, 32, 34, 35, 36, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 55, 57, 58, 59, 61, 62, 63, 66, 67, 68, 70, 71, 72, 74, 75, 76, 80, 81, 82, 84, 85, 86, 88, 89, 90, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 0

N. J. A. Sloane

nonn

3-adic valuation of (3n)!; cf. A054861.

Denominators of expansion of (1-x)^{-1/3} are 3^a(n). Numerators are in |A067622|.

Gary W. Adamson, in "Beyond Measure, A Guided Tour Through Nature, Myth and Number", by Jay Kappraff, World Scientific, 2002, p. 356.

T. D. Noe, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A004117, A001511, A051064, A055457.

Cf. A054861, A067080, A098844, A132027, A005187, A054899.

a004128 n = a004128_list !! (n-1)
a004128_list = scanl (+) 0 a051064_list
-- Reinhard Zumkeller, May 23 2013

[n + Valuation(Factorial(n), 3): n in [0..70]]; // Vincenzo Librandi, Jun 12 2019

A004128 := proc(n)
    A054861(3*n) ;
end proc:
seq(A004128(n),n=0..100) ; # R. J. Mathar, Nov 04 2017

Table[Total[NestWhileList[Floor[#/3] &, n, # > 0 &]], {n, 0, 70}] (* Birkas Gyorgy, May 20 2012 *)
A004128 = Log[3, CoefficientList[ Series[1/(1+x)^(1/3), {x, 0, 100}], x] // Denominator] (* Jean-François Alcover, Feb 19 2015 *)
Flatten[{0, Accumulate[Table[IntegerExponent[3*n, 3], {n, 1, 100}]]}] (* Vaclav Kotesovec, Oct 17 2019 *)

{a(n) = my(s, t=1); while(t<=n, s += n\t; t*=3);s}; /* Michael Somos, Feb 26 2004 */

a(n) = (3*n-sumdigits(n,3))/2; \\ Christian Krause, Jun 10 2025

def A007949(n):
    c = 0
    while not (a:=divmod(n,3))[1]:
        c += 1
        n = a[0]
    return c
def A004128(n): return n+sum(A007949(i) for i in range(3,n+1)) # Chai Wah Wu, Feb 28 2025

A004128 = lambda n: A004128(n//3) + n if n > 0 else 0
[A004128(n) for n in (0..70)]  # Peter Luschny, Nov 16 2012

A051064(n) = a(n+1) - a(n). - Alford Arnold, Jul 19 2000
a(n) = n + floor(n/3) + floor(n/9) + floor(n/27) + ... = n + a(floor(n/3)) = n + A054861(n) = A054861(3n) = (3*n - A053735(n))/2. - Henry Bottomley, May 01 2001
a(n) = Sum_{k>=0} floor(n/3^k). a(n) = Sum_{k=0..floor(log_3(n))} floor(n/3^k), n >= 1. - Hieronymus Fischer, Aug 14 2007
Recurrence: a(n) = n + a(floor(n/3)); a(3n) = 3*n + a(n); a(n*3^m) = 3*n*(3^m-1)/2 + a(n). - Hieronymus Fischer, Aug 14 2007
a(k*3^m) = k*(3^(m+1)-1)/2, 0 <= k < 3, m >= 0. - Hieronymus Fischer, Aug 14 2007
Asymptotic behavior: a(n) = (3/2)*n + O(log(n)), a(n+1) - a(n) = O(log(n)); this follows from the inequalities below. - Hieronymus Fischer, Aug 14 2007
a(n) <= (3n-1)/2; equality holds for powers of 3. - Hieronymus Fischer, Aug 14 2007
a(n) >= (3n-2)/2 - floor(log_3(n)); equality holds for n = 3^m - 1, m > 0. - Hieronymus Fischer, Aug 14 2007
Lim inf (3n/2 - a(n)) = 1/2, for n->oo. - Hieronymus Fischer, Aug 14 2007
Lim sup (3n/2 - log_3(n) - a(n)) = 0, for n->oo. - Hieronymus Fischer, Aug 14 2007
Lim sup (a(n+1) - a(n) - log_3(n)) = 1, for n->oo. - Hieronymus Fischer, Aug 14 2007
G.f.: (Sum_{k>=0} x^(3^k)/(1-x^(3^k)))/(1-x). - Hieronymus Fischer, Aug 14 2007
a(n) = Sum_{k>=0} A030341(n,k)*A003462(k+1). - Philippe Deléham, Oct 21 2011
a(n) ~ 3*n/2 - log(n)/(2*log(3)). - Vaclav Kotesovec, Oct 17 2019

Current definition suggested by Jason Earls, Jul 04 2001

A007417 If k appears, 3k does not.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 63, 64, 65, 67, 68, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 92, 94, 95, 97, 98, 99, 100

Offset: 1

N. J. A. Sloane and Simon Plouffe

The characteristic function of this sequence is given by A014578. - Philippe Deléham, Mar 21 2004
Numbers whose ternary representation ends in even number of zeros. - Philippe Deléham, Mar 25 2004
Numbers for which 3 is not an infinitary divisor. - Vladimir Shevelev, Mar 18 2013
Where odd terms occur in A051064. - Reinhard Zumkeller, May 23 2013

			From _Gary W. Adamson_, Mar 02 2010: (Start)
Given the following multiplication table: top row = "not multiples of 3", left column = powers of 3; we get:
   1   2   4   5   7   8   10   11   13
   3   6  12  15  21  24   30   33   39
   9  18  36  45  63  72   90   99  114
  27  54 108
  81
If rows are labeled (1, 2, 3, ...) then odd-indexed rows are in the set; but evens not. Examples: 9 is in the set since 3 is not, but 27 in row 4 can't be. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
Simon Plouffe, Email to N. J. A. Sloane, Jun. 1994
David Wakeham and David R. Wood, On multiplicative Sidon sets, INTEGERS, 13 (2013), #A26.
Index entries for 3-automatic sequences.

Complement of A145204. - Reinhard Zumkeller, Oct 04 2008

Cf. A007949, A014578 (characteristic function), A042948, A051064, A052330, A092400, A092401.

import Data.List (delete)
a007417 n = a007417_list !! (n-1)
a007417_list = s [1..] where
   s (x:xs) = x : s (delete (3*x) xs)

Select[ Range[100], (# // IntegerDigits[#, 3]& // Split // Last // Count[#, 0]& // EvenQ)&] (* Jean-François Alcover, Mar 01 2013, after Philippe Deléham *)
Select[Range[100], EvenQ@ IntegerExponent[#, 3] &] (* Michael De Vlieger, Sep 01 2020 *)

is(n) = { my(i = 0); while(n%3==0, n/=3; i++); i%2==0; } \\ Iain Fox, Nov 17 2017

is(n)=valuation(n,3)%2==0; \\ Joerg Arndt, Aug 08 2020

from sympy import integer_log
def A007417(n):
    def f(x): return n+x-sum(((m:=x//9**i)-2)//3+(m-1)//3+2 for i in range(integer_log(x,9)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 15 2025

Limit_{n->infinity} a(n)/n = 4/3. - Philippe Deléham, Mar 21 2004
Partial sums of A092400. Indices of even numbers in A007949. Indices of odd numbers in A051064. a(n) = A092401(2n-1). - Philippe Deléham, Mar 29 2004
{a(n)} = A052330({A042948(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Aug 31 2019

More terms from Philippe Deléham, Mar 29 2004

Typo corrected by Philippe Deléham, Apr 15 2010

A145204 Numbers whose representation in base 3 (A007089) ends in an odd number of zeros.

Original entry on oeis.org

0, 3, 6, 12, 15, 21, 24, 27, 30, 33, 39, 42, 48, 51, 54, 57, 60, 66, 69, 75, 78, 84, 87, 93, 96, 102, 105, 108, 111, 114, 120, 123, 129, 132, 135, 138, 141, 147, 150, 156, 159, 165, 168, 174, 177, 183, 186, 189, 192, 195, 201, 204, 210, 213, 216, 219, 222, 228, 231

Offset: 1

Reinhard Zumkeller, Oct 04 2008

nonn

Previous name: Complement of A007417.
Also numbers having infinitary divisor 3, or the same, having factor 3 in their Fermi-Dirac representation as product of distinct terms of A050376. - Vladimir Shevelev, Mar 18 2013
For n > 1: where even terms occur in A051064. - Reinhard Zumkeller, May 23 2013
If we exclude a(1) = 0, these are numbers whose squarefree part is divisible by 3, which can be partitioned into numbers whose squarefree part is congruent to 3 mod 9 (A055041) and 6 mod 9 (A055040) respectively. - Peter Munn, Jul 14 2020
The inclusion of 0 as a term might be viewed as a cultural preference: if we habitually wrote numbers enclosed in brackets and then used a null string of digits for zero, the natural number sequence in ternary would be [], [1], [2], [10], [11], [12], [20], ... . - Peter Munn, Aug 02 2020
The asymptotic density of this sequence is 1/4. - Amiram Eldar, Sep 20 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
Index entries for 3-automatic sequences.

Subsequence of A008585, A028983.
Subsequences: A016051, A055040, A055041, A329575.
Cf. A007089, A007417 (complement), A050376, A182581 (characteristic function).
Positions of 0s in A014578.
Excluding 0: the positions of odd numbers in A007949; equivalently, of even numbers in A051064; symmetric difference of A003159 and A036668.
Related to A042964 via A052330.
Related to A036554 via A064614.

a145204 n = a145204_list !! (n-1)
a145204_list = 0 : map (+ 1) (findIndices even a051064_list)
-- Reinhard Zumkeller, May 23 2013

isA145204 := proc(n) local d, c;
if n = 0 then return true fi;
d := A007089(n); c := 0;
while irem(d, 10) = 0 do c := c+1; d := iquo(d, 10) od;
type(c, odd) end:
select(isA145204, [$(0..231)]); # Peter Luschny, Aug 05 2020

Select[ Range[0, 235], (# // IntegerDigits[#, 3]& // Split // Last // Count[#, 0]& // OddQ)&] (* Jean-François Alcover, Mar 18 2013 *)
Join[{0}, Select[Range[235], OddQ @ IntegerExponent[#, 3] &]] (* Amiram Eldar, Sep 20 2020 *)

import numpy as np
def isA145204(n):
    if n == 0: return True
    c = 0
    d = int(np.base_repr(n, base = 3))
    while d % 10 == 0:
        c += 1
        d //= 10
    return c % 2 == 1
print([n for n in range(231) if isA145204(n)]) # Peter Luschny, Aug 05 2020

from sympy import integer_log
def A145204(n):
    if n == 1: return 0
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+sum(((m:=x//9**i)-2)//3+(m-1)//3+2 for i in range(integer_log(x,9)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 15 2025

a(n) = 3 * A007417(n-1) for n > 1.
A014578(a(n)) = 0.
For n > 1, A007949(a(n)) mod 2 = 1. [Edited by Peter Munn, Aug 02 2020]
{a(n) : n >= 2} = {A052330(A042964(k)) : k >= 1} = {A064614(A036554(k)) : k >= 1}. - Peter Munn, Aug 31 2019 and Dec 06 2020

New name using a comment of Vladimir Shevelev by Peter Luschny, Aug 05 2020

A055457 5^a(n) exactly divides 5n. Or, 5-adic valuation of 5n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2

Offset: 1

Alford Arnold, Jun 25 2000

More generally, consider the sequence defined by p^a(n) exactly divides p*n. For p = 3 we have A051064 and for p = 2 we have A001511.

The number of powers of 5 that divide n. - Amiram Eldar, Mar 29 2025

			a(5) = 2 since 5^2 exactly divides 5 times 5;
a(25) = 3 since 5^3 exactly divides 5 times 25;
a(125) = 4 since 5^4 exactly divides 5 times 125.

T. D. Noe, Table of n, a(n) for n=1..1000
Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (The A indices in P at equations 1' and 2' for p=5.)

Cf. A007949, A112765, A191610 (partial sums).

Cf. A001511, A051064, A115362, A373216, A373217.

seq(padic:-ordp(5*n,5), n=1..1000); # Robert Israel, Dec 07 2015

max = 1000; s = (1/x)*Sum[x^(5^k)/(1-x^5^k), {k, 0, Log[5, max] // Ceiling }] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 04 2015 *)
Table[IntegerExponent[n, 5] + 1, {n, 1, 100}] (* Amiram Eldar, Sep 21 2020 *)

a(n)=-sumdiv(n,d,moebius(5*d)*numdiv(n/d)) \\ Benoit Cloitre, Jun 21 2007

a(n)=valuation(5*n,5) \\ Anders Hellström, Dec 04 2015

def A055457(n):
    c = 1
    while not (a:=divmod(n,5))[1]:
        c += 1
        n = a[0]
    return c # Chai Wah Wu, Feb 28 2025

G.f.: Sum_{k>=0} x^(5^k)/(1-x^5^k). - Ralf Stephan, Apr 12 2002
Multiplicative with a(p^e) = e+1 if p = 5, 1 otherwise.
a(n) = -Sum_{d|n} mu(5d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
Dirichlet g.f.: zeta(s)/(1-1/5^s). - R. J. Mathar, Feb 09 2011
a(n) = A112765(5n). - R. J. Mathar, Jul 17 2012
a(5n) = 1 + a(n).  a(5n+k) = 1 for k = 1..4. - Robert Israel, Dec 07 2015
G.f. satisfies A(x^5) = A(x) - x/(1-x). - Robert Israel, Dec 08 2015
a(n) = A112765(n) + 1. - Amiram Eldar, Sep 21 2020
Sum_{k=1..n} a(k) ~ 5*n/4. - Vaclav Kotesovec, Sep 21 2020
G.f.: Sum_{i>=1, j>=0} x^(i*5^j). - Seiichi Manyama, Mar 23 2025

A096346 Complement of A004128.

Original entry on oeis.org

3, 7, 11, 12, 16, 20, 24, 25, 29, 33, 37, 38, 39, 43, 47, 51, 52, 56, 60, 64, 65, 69, 73, 77, 78, 79, 83, 87, 91, 92, 96, 100, 104, 105, 109, 113, 117, 118, 119, 120, 124, 128, 132, 133, 137, 141, 145, 146, 150, 154, 158, 159, 160, 164, 168, 172, 173, 177, 181, 185

Offset: 0

Alford Arnold, Aug 04 2004

Shape sequence for A055938 is A001511; shape sequence for a(n) is A051064; A001511, A051064 and A055457 are p-adic valuations for p = 2, 3 & 5.

Also n! never ends in this many 0's in bases 3 and 6. - Carl R. White, Jan 21 2008

			A004128 begins 0 1 2 4 5 6 8 9 10 13 14 15 ... therefore a(n) begins 3 7 11 12 16 20 24 25 ...

Cf. A055457, A086343.

b:=n->sum(floor(3*n/3^k),k=1..n): {seq(n,n=0..222)} minus {seq(b(n),n=0..150)}; # Emeric Deutsch, Dec 09 2004

A004128 = Log[3, CoefficientList[ Series[ 1/(1+x)^(1/3), {x, 0, 200}], x] // Denominator]; A096346 = Complement[ Range[A004128 // Last], A004128] (* Jean-François Alcover, Feb 19 2015 *)

More terms from Emeric Deutsch, Dec 09 2004

A254046 Column index of n in A191450: a(3n) = 1, a(3n+1) = 1, a(3n+2) = 1 + a(n+1).

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 6, 1, 1, 2, 1, 1, 2

Offset: 1

Antti Karttunen, Jan 24 2015

Equally, the row index of n in A254051.

a(n) is the 3-adic valuation of A087289(n-1). - Fred Daniel Kline, Jan 11 2017

Antti Karttunen, Table of n, a(n) for n = 1..9842

One more than A253786.
Cf. A253887 (the corresponding row index).
Cf. A191450, A254047, A254051, A254052.
Odd bisection of A051064.
Cf. A007949, A087289.

With[{nmax=200},IntegerExponent[6Range[nmax]-3,3]] (* Paolo Xausa, Nov 10 2023 *)

a(3n) = 1, a(3n+1) = 1, a(3n+2) = 1 + a(n+1).
a(n) = A253786(n) + 1.
a(n) = A253786(3n-1). - Cyril Damamme, Aug 04 2015
a(n) = A051064(2n-1), i.e., the 3-adic valuation of 6n-3. - Cyril Damamme, Aug 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Nov 16 2023

A320107 a(n) = A001227(A252463(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 4, 2, 1, 2, 2, 4, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 4, 2, 2, 2, 4, 2, 3, 2, 2, 3, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 2, 2, 2, 6, 4, 2, 2, 2, 4, 2, 2, 3, 2, 3, 2, 4, 2, 2, 4

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Records 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, ... occur at n = 1, 5, 18, 30, 90, 210, 450, 630, 1890, 3150, 5670, 6930, 20790, 34650, 62370, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001227, A005940, A051064, A055457, A252463, A320106 (Möbius transform).

Cf. also A320111 and A319699, A319700, A319703, A319989.

A001227(n) = numdiv(n>>valuation(n, 2));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A320107(n) = A001227(A252463(n));

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A320107(n) = if(n<=2, 1, my(p=A252463(n)); if(!(p%2), A320107(p), A051064(p)*A320107(p)));

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A055457(n) = if(n<1, 0, 1+valuation(n, 5));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A320107(n) = if(n<=2, 1, if(!(n%4), A320107(n/2), if(2==(n%4), A051064(n)*A320107(n/2), if(!(n%3), A320107(A064989(n)), A055457(n)*A320107(A064989(n))))));

a(n) = A001227(A252463(n)).
a(1) = a(2) = 1; for n > 2, a(n) = a(n/2) when n == 0 mod 4, a(n) = A051064(n) * a(n/2) when n == 2 mod 4, a(n) = a(A064989(n)), when n == 3 mod 6, otherwise a(n) = A055457(n) * a(A064989(n)).
For n > 2, let p = A252463(n). If p is even, then a(n) = a(p), if p is odd, then a(n) = A051064(p) * a(p).

A087088 Positive ruler-type fractal sequence with 1's in every third position.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 1, 5, 3, 1, 2, 6, 1, 4, 2, 1, 3, 7, 1, 2, 5, 1, 3, 2, 1, 4, 8, 1, 2, 3, 1, 6, 2, 1, 4, 3, 1, 2, 5, 1, 9, 2, 1, 3, 4, 1, 2, 7, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 6, 2, 1, 10, 3, 1, 2, 4, 1, 5, 2, 1, 3, 8, 1, 2, 4, 1, 3, 2, 1, 6, 5, 1, 2, 3, 1, 4, 2, 1, 7, 3, 1, 2, 11, 1, 4, 2, 1, 3, 5, 1, 2, 6, 1, 3, 2

Offset: 1

Enrico T. Federighi (rico125162(AT)aol.com), Aug 08 2003

If all the terms in the sequence are reduced by one and then all zeros are removed, the result is the same as the original sequence.
From Benoit Cloitre, Mar 07 2009: (Start)
To construct the sequence:
Step 1: start from a sequence of 1's and leave two undefined places between every pair of 1's giving: 1,(),(),1,(),(),1,(),(),1,(),(),1,(),(),1,...
Step 2: replace the first undefined place with a 2 and henceforth leave two undefined places between two 2's giving: 1,2,(),1,(),2,1,(),(),1,2,(),1,(),2,1,...
Step 3: replace the first undefined place with a 3 and henceforth leave two undefined places between two 3's giving: 1,2,3,1,(),2,1,(),3,1,2,(),1,(),2,1,...
Step 4: replace the first undefined place with a 4 and leave 2 undefined places between two 4's giving: 1,2,3,1,4,2,1,(),3,1,2,(),1,4,2,1,... Iterating the process indefinitely yields the sequence: 1,2,3,1,4,2,1,5,3,1,2,6,1,4,2,1,... (End)
From Peter Munn, Jul 10 2020: (Start)
For k >= 1, the number k occurs in a pattern with fundamental period 3^k, and with points of mirror symmetry at intervals of (3^k)/2. Those points have an extrapolated common origin (for k >= 1) at an offset 1.5 to the left of the sequence's initial "1". The snake format illustration in the example section may be useful for visualizing this.
(End)
For k >= 1, k first occurs at position A061419(k) and its k-th occurrence is at position A083045(k-1). - Peter Munn, Aug 23 2020
(a(n)) is the unique fixed point of the two-block substitution a,b -> 1,a+1,b+1, where a,b are natural numbers. - Michel Dekking, Sep 26 2022

			From _Peter Munn_, Jul 03 2020: (Start)
Listing the terms in a snake format (with period 27) illustrates periodic and mirror symmetries. Horizontal lines mark points of mirror symmetry for 3's. Vertical lines mark further points of mirror symmetry for 2's. 79 terms are shown. (Referred to the extrapolated common origin of periodic mirror symmetry, the initial term is at offset 1.5 and the last shown is at offset 79.5 = 3^4 - 1.5.) Observe also mirror symmetry of 4's (seen vertically).
    1  2  3  1  4  2  1  5   3  1  2  6
             |             |            1 --
    1  2  3  1  5  2  1  7   3  1  2  4
_ 4
  8
    1  2  3  1  6  2  1  4   3  1  2  5
             |             |            1 --
    1  2  3  1  7  2  1  4   3  1  2  9
_ 5
  4
    1  2  3  1  6  2  1 10   3  1  2  4
             |             |            1 --
    1  2  3  1  4  2  1  8   3  1  2  5
(End)
From _Peter Munn_, Aug 22 2020: (Start)
The start of the sequence is shown below in conjunction with related sequences, aligning their points of mirror symmetry. The longer, and shorter, vertical lines indicate points of mirror symmetry for terms valued less than 4, and less than 3, respectively. Note each term of A051064 is the minimum of two terms displayed nearest below it, and each term of A254046 is the minimum of the two terms displayed diagonally above it.
        |                          |                          |
A051064:| 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1 4 1 1 2
        |        |        |        |        |        |        |
[a(n)]: |  1 2 3 1 4 2 1 5 3 1 2 6 1 4 2 1 3 7 1 2 5 1 3 2 1 4 8 1 2 3
        |        |        |        |        |        |        |
A254046:|1 2 1 1 3 1 1 2 1 1 2 1 1 4 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1
        |                          |                          |
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A001511, A244040.
Sequences with equivalent symmetries: A051064, A254046.
Records are given by A061419: a(A061419(n))=n.
Essentially the odd bisection of A335933.
Sequence with similar definition: A087165.
Ordinal transform of A163491, with which this sequence has a joint relationship to A083044, A083045.
See also the comment in A024629.

A163491[n_] := A163491[n] = If[Mod[n, 3]==1, (n+2)/3, A163491[Floor[2n/3]]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A163491[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

a(n) = 1 when n == 1 (mod 3), otherwise a(n) = a(n-ceiling(n/3)) + 1.
a(n) = 3 + A244040(3*(n-1)) - A244040(3*n). - Tom Edgar and James Van Alstine, Aug 04 2014
From Peter Munn, Aug 22 2020: (Start)
For m >= 0, a(3*m+1) = 1; a(3*m+2) = a(2*m+1) + 1; a(3*m+3) = a(2*m+2) + 1.
For n >= 1, the following identities hold.
a(n) = A335933(2*n+1).
A083044(A163491(n) - 1, a(n) - 1) = n.
A051064(n+1) = min(a(n), a(n+1)).
A254046(n+2) = min(a(n), a(n+2)). (End)

More terms from Paul D. Hanna, Aug 21 2003
Offset changed by M. F. Hasler (following remarks by Peter Munn), Jul 13 2020
Thanks to Allan C. Wechsler for suggesting the new name. - N. J. A. Sloane, Jul 14 2020

A070529 Number of divisors of repunit 111...111 (with n digits).

Original entry on oeis.org

1, 2, 4, 4, 4, 32, 4, 16, 12, 16, 4, 128, 8, 16, 64, 64, 4, 384, 2, 128, 128, 96, 2, 1024, 32, 64, 64, 256, 32, 8192, 8, 2048, 64, 64, 128, 3072, 8, 8, 64, 2048, 16, 24576, 16, 1536, 768, 64, 4, 8192, 16, 1024, 256, 512, 16, 8192, 256, 4096

Offset: 1

Henry Bottomley, May 02 2002

			a(9) = 12 since the divisors of 111111111 are 1, 3, 9, 37, 111, 333, 333667, 1001001, 3003003, 12345679, 37037037, 111111111.

Table of n, a(n) for n = 1..352

Cf. A000005, A002275, A070528, A051064, A004023, A046413, A003020, A095370, A176973, A046053, A046412, A102380.

a(n) = numdiv(10^n\9); \\ Michel Marcus, Sep 09 2015

a(n) = A000005(A002275(n)).
a(n) = A070528(n)*A051064(n)/(A051064(n)+2).
a(A004023(n)) = 2. - Michel Marcus, Sep 09 2015
a(A046413(n)) = 4. - Bruno Berselli, Sep 09 2015

Terms to a(280) in b-file from Hans Havermann, Aug 20 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) ib b-file from Max Alekseyev, May 04 2022

A327625 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^2.

Original entry on oeis.org

1, 2, 4, 4, 5, 8, 7, 8, 13, 10, 11, 16, 13, 14, 20, 16, 17, 26, 19, 20, 28, 22, 23, 32, 25, 26, 40, 28, 29, 40, 31, 32, 44, 34, 35, 52, 37, 38, 52, 40, 41, 56, 43, 44, 65, 46, 47, 64, 49, 50, 68, 52, 53, 80, 55, 56, 76, 58, 59, 80, 61, 62, 91, 64, 65, 88, 67, 68, 92, 70

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

Sum of divisors d of n such that n/d is power of 3.

Inverse Moebius transform of A195459.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A000244, A001651 (fixed points), A051064, A129527, A195459.

[(1/2)*&+[EulerPhi(3*d) :d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Sep 19 2019

nmax = 70; CoefficientList[Series[Sum[x^(3^k)/(1 - x^(3^k))^2, {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[Log[3, n/#]] &]; Table[a[n], {n, 1, 70}]
a[n_] := 1/2 Sum[EulerPhi[3 d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

A327625(n) = (n+sumdiv(n,d,my(b=0); if(isprimepower(n/d,&b)&&(3==b),d,0))); \\ Antti Karttunen, Sep 19 2019

G.f. A(x) satisfies: A(x) = A(x^3) + x/(1 - x)^2.
G.f.: Sum_{k>=1} phi(3*k) * x^k / (2 * (1 - x^k)), where phi = A000010.
a(n) = (1/2) * Sum_{d|n} phi(3*d).
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(3^e) = (3^(e+1)-1)/2, and a(p^e) = p^e for p != 3.
Sum_{k=1..n} a(k) ~ (9/16) * n^2. (End)
Dirichlet g.f.: zeta(s-1)*(1+1/(3^s-1)). - Amiram Eldar, Dec 17 2022

cp's OEIS Frontend

A004128 a(n) = Sum_{k=1..n} floor(3*n/3^k).

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A007417 If k appears, 3k does not.

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A145204 Numbers whose representation in base 3 (A007089) ends in an odd number of zeros.

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A055457 5^a(n) exactly divides 5n. Or, 5-adic valuation of 5n.

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A096346 Complement of A004128.

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A254046 Column index of n in A191450: a(3n) = 1, a(3n+1) = 1, a(3n+2) = 1 + a(n+1).

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